。
She somehow managed to back through the narrow road.
Literal
She [topic-は] that narrow road [object-を] somehow back-with passed-through.
何とか conveys 'somehow / by some means / barely' — usually with a sense of effort or difficulty overcome. Here it amplifies the 'just barely' feeling of squeezing a car through a tight spot in reverse. バック is an English loanword used as both a noun and a verbal element ('to back up' a vehicle); で marks it as the means or method. The を on 道 is the path-を, marking the road as the route traversed by 通り抜ける ('pass all the way through') — common with motion verbs like 通る, 歩く, 渡る.